The pressure in the pipe near the discharge of a reciprocati

The pressure in the pipe near the discharge of a reciprocating pump fluctuates according to p = 200+40sin(8t) where t is in seconds and p is kPa. If The fluid speed in the pipe is 5m/s. what is the maximum rate of change of pressure experienced by a fluid particle?

Solution

p = 200 + 40*sin 8t

u = 5 m/sec, v = 0 and w = 0

given that p = p(t)

DP/Dt = dp/dt + u*dp/dx + v*dp/dy + w*dp/dz

using above information

Since dp/dx = 0 so value of u does not matter.

DP/Dt = dp/dt + 0 + 0 + 0

DP/Dt = d(200 + 40*sin 8t)/dt

DP/Dt = (320*cos 8t) kPa

DP/Dt)max when cos 8t = max = 1

DP/Dt)max = 320 kPa